Re: Pure-cardinal approach *is* possible! (was: Mathematical concepts)

In article <REM-2005sep01-003@xxxxxxxxx>,
Robert Maas, see http://tinyurl.com/uh3t <rem642b@xxxxxxxxx> wrote:
>> From: hru...@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
>> it is necessary to separate the understanding of the concepts from
>> the ability to produce proofs.

>I agree. But how would you propose testing whether a child really does
>understand a concept, before declaring this lesson finished and moving
>on to the next lesson that teaches a different concept?

>> >Which is one reason why I don't advocate even mentionning anything
>> >infinite, not even the set of natural numbers, on the first day of
>> >introductory pre-school cardinality and ordinality.

>> Then they will have difficulty later.

>That's fine with me.

Unlearning is the HARDEST part.

Your alternative seems to be to push an entire
>graduate-level foundations course

It is FAR less than that. Also, one can have rigor
without having completeness; concepts can be understood
without going through all the proofs.

at a 3-yr-old the very first day they
>ever hear of the concept of cardinal numbers, so they get so frustrated
>they give up already and never want to ever try any mathematics of any
>kind ever again.

This is highly questionable if done correctly. Right
now, they come out of high school UNABLE to learn
mathematics. It is not necessary to do everything at
once, but do things in such a way that misconceptions
do not occur.

By my method, the first lesson is easy, the second
>adds to it, etc. until after a few lessons the graduate-level stuff is
>merely difficult/interesting, not impossible/frustrating.

>> > HUNDREDS TENS ONES
>> This is introducing base 10 too early.

>Third grade is too early??? You gotta be kidding!

You are assuming too much and too little ability.

>There's plenty of time for the 3-yr-old to learn cardinal and ordinal
>number concepts, and Peano's axioms, etc., and for the 4-yr-old to
>learn how to construct proofs, and for the 5-yr-old to solve previously
>unsolved problems, before finally getting base ten.

I never assumed that much, but somewhat in that nature.
The game of WFF 'N PROOF is about well-formed formulas
(expressions) and proofs.

However, it is only at the PhD level that one solves
previously unsolved problems.

>> The use of the different columns can be developed from the
>> abstract standpoint than using different symbols in the
>> different places, or different marks for each power of 10.

>Why do you propose teaching something the child will need to un-learn
>later?? With popsicle sticks, each of which looks like this: | you can
>have a number that looks like this ||||||||||||||||||||||||||||||||||
>and then break it up into meaningful pieces any way you want. For work
>on computers, it might be best to group the sticks by |||| and then to
>group those bunches by |||| |||| |||| |||| and then group those bunches
>of bunches by |||| |||| |||| |||| |||| |||| |||| ||||
>|||| |||| |||| |||| |||| |||| |||| |||| etc. So that random-size
>clump I cited earlier would be grouped and laid out this way:
>|||| |||| |||| |||| |||| |||| |||| |||| ||
>(It wouldn't really look like that, because each group of |||| would be
>clumped together with a rubber band around it, then each next-level
>likewise, so it'd be easy to recognize the distinctly different sizes
>of standard groups.) Then the teacher could mention that the
>traditional way (centuries before computers were invented) is more like
>this: ||||| then ||||| ||||| then
>||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||||
>then
>||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||||
>||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||||
>etc. So that random-size clump I cited earlier would now be grouped and
>laid out this way: ||||| ||||| ||||| ||||| ||||| ||||| ||||
>Now we have the foundation for oldstyle tallies, newstyle tallies,
>oldstyle Roman numerals, and Arabic numerals in a positional system.

This looks like teaching base 4. It is much the way I would
teach the idea of base. However, other than the Sumerian
invention of base 60 and the Mayan of base 20, it seems that
only base 10 was ever used, and until the Indian numbers, these
base 10 systems had different symbols in different places.

I would use tick marks instead of popsicle sticks, but it would
not be much different.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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